Another open question is whether the distance scaling of Theorems REF 
and REF  can be extended to apply also to \(k\)  to some degree.
Improving the fault-tolerant gadgets or using specially designed base
codes seem like obvious avenues to try to improve on our codes.
Conversely, extending the existing upper
bounds by Bravyi [1]} to \(D>2\) , as well as extending the bound from
Bravyi, Poulin, and Terhal [2]} to subsystem codes would be also
be interesting. If the bound in Eq. (REF ) extended to subsystem codes,
then the scaling in Theorem REF  would be tight; alternatively, it is
possible that spatially local subsystem codes can achieve asymptotically better parameters
than spatially local stabilizer codes.
